Denote the Teichmüller curve corresponding to the lattice \(\operatorname{Stab}({\widetilde{\phi }_n})\)  by \((\mathchoice{\chi }{}{}{}\)\(\chi \) n, n)\(. An algebraic description of the map\begin{equation}\phi _n \colon \mathchoice{\chi }{}{}{}\end{equation}{\chi }\) n Mg,

given in [1]}, [2]}, and [3]}, is as follows. Let \(\widetilde{\mathchoice{\chi }{}{}{}}{\chi }\) n := (1-{n})\(, where \) n\( denotes the set of \) n\(^{\prime }th roots of unity. Let \) t\( be the restriction of the global rational coordinate on \) 1\( to \)\(\chi \)\(\chi \) n\( and let \) Dn Aut(\(\chi \)\(\chi \) n)\( denote the dihedral group generated by \) t n2t\( and \) t 1/ t\( where \) n := (2-1n)\(. For all \) t \(\chi \)\(\chi \) n\( let \) Ft\( be the unique compact hyperelliptic Riemann surface associated with the polynomial{\begin{@align*}{1}{-1}y^2= \prod _{i=0}^n (x - \zeta _n^{i} - t\zeta _n^{-i}).\end{@align*}}It turns out that \)\(\chi \)\(\chi \) n \(\chi \)\(\chi \) n/Dn\( and the isomorphism class of \) Yt\( depends only on \) [t] \(\chi \)\(\chi \) /D\(. The map (\ref {phin}) is then simply given, after realizing \)\(\chi \)\(\chi \) n\( as the quotient \)\(\chi \)\(\chi \) n/Dn\(, by \) n([t])= [Ft] Mg\(.\)
